
Case 1. \({\bf {\bf (C_1)}}\) , \({\bf (H_B^1)}\) , \({\bf (H_\chi ^j)}\)  and \({\bf (H_\psi ^k)}\)  hold.
Then
\(\chi (u_{R},-u_{R})+\psi ( \lambda _{R},- \lambda _{R})\ge -\Vert f\Vert _{X^\ast }\Vert u_{R}\Vert _{X}-\Vert g\Vert _{Y^\ast }\Vert \lambda _R\Vert _{Y}+ B(u_R,0_Y)-B(0_X,\lambda _R).\) 
Since every convex lower continuous functional is bounded below by an affine function (see, e.g., Brezis [1]}), there exist \(f_1\in X^\ast \) , \(g_1\in Y^\ast \)  and \(c_1,c_2\in \mathbb {R}\)  such that
\(B(u_R,0_Y)\ge \langle f_1,u_R\rangle +c_1\ge -\Vert f_1\Vert _{X^\ast }\Vert u_R\Vert _X+c_1,\) 
and
\(-B(0_X,\lambda _R)\ge \langle g_1,\lambda _R\rangle +c_2\ge -\Vert g_1\Vert _{Y^\ast }\Vert \lambda _R\Vert _Y+c_2.\) 
Thus,
\(\frac{\chi (u_R,-u_R)+\psi (\lambda _R,-\lambda _R)}{\sqrt{ \Vert u_R\Vert _X^2+\Vert \lambda _R\Vert _Y^2}}\ge \frac{-c_3\Vert u_R\Vert _X-c_4\Vert \lambda _R\Vert _Y+c_1+c_2}{\sqrt{ \Vert u_R\Vert _X^2+\Vert \lambda _R\Vert _Y^2}},\) 
where \(c_3:=\Vert f\Vert _{X^\ast }+\Vert f_1\Vert _{X^\ast }>0\)  and \(c_4:=\Vert g\Vert _{Y^\ast }+\Vert g_1\Vert _{Y^\ast }>0\) .
Letting \(R\rightarrow \infty \)  we reach a contradiction as the left-hand side term tends to \(-\infty \) , while the right-hand side term is bounded (keep in mind we assumed \(\max \lbrace \Vert u_R\Vert _X,\Vert \lambda _R\Vert _Y\rbrace =R\) ).

Case 2. \({\bf {\bf (C_2)}}\) , \({\bf (H_B^i)}\) , \({\bf (H_\chi ^j)}\)  and \({\bf (H_\psi ^k)}\)  hold.
Using (REF ) we get
\(\frac{B(0_X,\lambda _R)-B(u_R,0_Y)}{\sqrt{\Vert u_R\Vert _X^2+\Vert  \lambda _R \Vert _Y^2 }^{\max \lbrace p,q\rbrace }}\ge -\frac{\Vert f\Vert _{X^\ast } \Vert u_R\Vert _X+m_\chi \Vert u_R\Vert _X^p+\Vert g\Vert _{Y^\ast } \Vert \lambda _R\Vert _Y+m_\psi \Vert \lambda _R\Vert _X^q}{ \sqrt{\Vert u\Vert _X^2+\Vert \lambda \Vert _Y^2 }^{\max \lbrace p,q\rbrace } },\) 
and a contradiction is reached by letting \(R\rightarrow \infty \) .

